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Date: 19-11-2021
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A vector field is a map that assigns each
a vector
. Several vector fields are illustrated above. A vector field is uniquely specified by giving its divergence and curl within a region and its normal component over the boundary, a result known as Helmholtz's theorem (Arfken 1985, p. 79).
Vector fields can be plotted in the Wolfram Language using VectorPlot[f, x, xmin, xmax
,
y, ymin, ymax
].
Flows are generated by vector fields and vice versa. A vector field is a tangent bundle section of its tangent bundle.
REFERENCES:
Arfken, G. "Vector Analysis." Ch. 1 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 1-84, 1985.
Gray, A. "Vector Fields on " and "Derivatives of Vector Fields on
." §11.4 and 11.5 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 255-258, 1997.
Morse, P. M. and Feshbach, H. "Vector Fields." §1.2 in Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 8-21, 1953.
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